Consider a vectorfield on the surface of a sphere, which is a (continuously differentiable) function representing the wind on earth. Now, let's say I want to calculate the path of an airplane such that the path minimizes its fuel consumption.
So, I am looking for a function minimizing a functional. The functional is basically the fuel cost of the path, so the time it takes to fly the path times some constant. It sounds to me as a problem where the Euler-Lagrange equations are usable, but after a quick try the equations tend to become extremely difficult (like in more then 3 pages long).
Is there some way to avoid these difficulties? The usage of numerical methods is also perfectly fine.